Self-affine quadrangles
Christian Richter, Felix Zimmermann
TL;DR
This work provides a complete classification of 3-self-affine convex quadrangles by translating classical gc-parameterizations into a natural Q[x,y] framework and solving the resulting algebraic constraints. It separates glass-cut and non-glass-cut dissections, yielding five principal families (T,A,B1,B2,C) plus 13 singular parameter cases that fully describe all 3-self-affine convex quadrangles. Extending beyond convexity, the paper proves the existence of n-self-affine non-convex quadrangles for every n≥3 (but not for n=2), via explicit constructions and invariance arguments. The results reduce the search for 4-self-affine convex quadrangles to future work on that open case, and they illustrate the broader geometric landscape of self-affinity in planar polygons with both convex and non-convex examples.
Abstract
A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles are trapezoids. Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$. In addition, we show that there are $n$-self-affine non-convex quadrangles for all $n \ge 3$, but not for $n=2$.